The optimal packing of congruent regular pentagons in the plane

The densest packing of congruent circles in the plane is the familiar hexagonal arrangement in which every circle touches six others.  This problem was solved many decades ago.  The next packing problem in the plane to consider is that of finding the densest packing of congruent regular polygons.   Congruent equilateral triangles tile the plane.  So do squares and regular hexagons.  Tilings have density one so the packing problem is trivial in these cases.  What about regular pentagons?  This talk will present work with Woden Kusner on determining the densest packing of regular pentagons.  This research is computer-assisted. Some of our algorithms will be described.